\documentclass{article}
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\title{Six-state clock model analysis}
\author{Vladimir Iglovikov}
\begin{document}
\maketitle
\section{Introduction} % (fold)
\label{sec:introduction}
In this paper I'll try to summarize different approaches to analyze KT transitions and add some of my bright ideas to create some consistent text 
describing methods that later will be applied to the Ising model in the transverse field.
% section introduction (end)
\section{Clock model, description} % (fold)
\label{sec:clock_model_description}
Clock model is described by the Hamiltonian
\begin{equation}
	H = -J \sum_{\langle i,j \rangle} \cos \left[ 2 \pi (n_i - n_j) / q \right]
\end{equation}
where $J$ - coupling constant, $q$ - integer. $n_i = 1,2,\ldots, q-1$.  I'll study $q =6$ model. I impose periodic boundary conditions and perform simulations. I perform $5 \cdot 10^3 \tau_C$ Monte Carlo steps (MCS) for equilibration and $10^5 \tau_C$ for averaging, where $\tau_C$ is defined as:

\begin{equation}
	\tau_C = 1/2 + \sum_{t=1}^N C(t) (1 - t / N)
\end{equation}
where $C(t)$ - autocorrelation function. $N$ - big integer. In theory should be $\infty$. I use 1000.

Quantities, that I measure in my simulation:
\begin{gather}
\tau_C\\
\langle E \rangle\\
C = L^2 \frac {\langle E^2 \rangle - \langle E \rangle^2} {T^2}\\
\langle |M| \rangle \\
\langle |M^2| \rangle\\
\langle |M^4| \rangle\\
X = L^2 \frac {\langle |M|^2 \rangle - \langle |M| \rangle^2} {T}
\end{gather}
where $E$ - energy, $C$ - specific heat, $|M|$ - absolute value of the total magnetization, $X$ - susceptibility

\section{How can we roughly estimate critical point/points} % (fold)
\subsection{Autocorrelation length}
At the critical point correlation length diverges, and correlation time has maximum. See Figure~\ref{fig:figure1}.
\begin{figure}[ht]
	\begin{center}
		\includegraphics[scale = 0.6]{graphs/q6_tau.png}
	\end{center}
	\caption{Correlation time $\tau_C$ vs temperature $T$}
	\label{fig:figure1}
\end{figure}
We can see that there are two peaks, so even we do not know nature of the critical point/points we can suspect that temperatures around $T = 0.63$ and $T = 1.07$ can be interesting.
\subsection{Binder cumulant} % (fold)
Binder cumulant is quantity defined as
\begin{equation}
	U_L = 1 - \frac {\langle m^4 \rangle} {3 \langle m^2 \rangle^2}
\end{equation}

At the critical point $U_L = U_{L'}=U_{\infty}$. This means, that if $U_L \ne U_{L'}$ it is not critical point. If $U_L = U_{L'}$ it may be critical, may be not. 

If we look at Figure~\ref{fig:figure2} we will see that above $T = 0.95$ there are no critical points.
\begin{figure}[ht]
	\begin{center}
		\includegraphics[scale = 0.5]{graphs/q6_cumulant.png}
	\end{center}
	\caption{Binder cumulant vs Temperature $T$ for different lattice sizes}
	\label{fig:figure2}
\end{figure}
Another thing that we can get from this graph is that shape of the Figure~\ref{fig:figure2} is different from a typical Binder cumulant of the second order phase transition. So we can suspect KT transition.
\label{sub:binder_cumulant}
% subsection binder_cumulant (end)
\subsection{$V_L$} % (fold)
At the KT critical point 
\begin{gather}
m_L L^{\eta/2} = Y_1(\xi/L)\\
X_L L^{2 - \eta} = Y_2(\xi/L)
\end{gather}
So quantity $V_L$ that I define as
\begin{equation}
	V_L \equiv L^2 \frac {m_L^2} {X_L}
\end{equation}
behaves as 
\begin{equation}
	V_L = Y_3(\xi/L)
\end{equation}
So behavior of $V_L$ is similar to the Binder cumulant behavior, but works only for the KT phase. $V_L \ne V_{L'}$ means that it is not KT critical point.
\begin{figure}[ht]
	\begin{center}
		\includegraphics[scale = 0.38]{graphs/q6_VL.png}
	\end{center}
	\caption{$V_L$ vs $T$, for different lattice sizes. If lines do not collapse, it is not KT critical point/points}
	\label{fig:figure3}
\end{figure}
If we look at Figure~\ref{fig:figure3}, we will see that there is no KT type critical point below $T = 0.65$.
\subsection{Magnetization} % (fold)
At KT critical point 
\begin{equation}
	m_L \sim L^{-\eta / 2}
\end{equation}
So if I plot $\ln (m_L)$ vs $\ln(L)$ in the KT phase this graph will be straight line. 

\begin{figure}[th]
	\begin{center}
		\includegraphics[scale = 0.37]{graphs/q6_ML.png}
	\end{center}
	\caption{$\ln(m_L)$ vs $\ln L$. If line is not straight - not in the KT phase}
	\label{fig:figure7}
\end{figure}
It is clear from Figure~\ref{fig:figure7} that $T = 1$, $T = 1.05$ are not in the KT phase. To tell something about other temperatures we need to do simulations on bigger lattice sizes.

Conclusion of this section is that rough analysis tells us that there is no second order phase transition. But probably we have extended KT phase within temperature range $T \in [0.65:0.95]$
\section{More accurate measurements} % (fold)
\subsection{Susceptibility}
At the KT critical point correlation length diverges as
\begin{equation}
\label{eq:xi}
\xi \sim \exp(a t^{-0.5})	
\end{equation}
On the final lattice at critical point $\xi = L$ and \eqref{eq:xi} gives us for the finite lattice
\begin{equation}
\label{eq:T_1C}
	T_C(L) = T_C(\infty) + \frac {T_C a^2} {\ln(L)^2}
\end{equation}
where $T_C(L)$ is position of the maximum of the susceptibility for the finite lattice, $T_C$ - critical temperature.
\begin{figure}[th]
	\begin{center}
		\includegraphics[scale = 0.6]{graphs/q6_X.png}
	\end{center}
	\caption{Susceptibility vs temperature}
	\label{fig:figure4}
\end{figure}
\begin{figure}[ht]
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{graphs/q6_TC1.png}
	\caption{Position of the first peak of the susceptibility vs $\ln(L)^{-2}$}
	\label{fig:figure5}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{graphs/q6_TC2.png}
\caption{Position of the second peak of the susceptibility vs $\ln(L)^{-2}$}
	\label{fig:figure6}
%\caption{data collapse of $X$}
\end{subfigure}	
\end{figure}
From Figure~\ref{fig:figure4} we can see that susceptibility has two peaks. I plot position of the susceptibility maximum versus $(\ln L)^{-2}$.
Intersection with vertical axis gives us critical temperature. $T_2 = 0.94$, $T_1 = 0.66$.

\section{Conclusions}
In this text I've used methods suggested in works \cite{Batrouni, clock} plus some of my bright ideas to analyze critical behavior of the six state clock model. I've got extended KT phase between temperatures $T_1 = 0.66$, $T_2 = 0.94$.
Results of the work \cite{clock} $T_1 = 0.68$, $T_2 = 0.92$. Now  I am planning to use this methods to analyze Antiferromagnetic Ising Model in the transverse field.

\begin{thebibliography}{99}
	\bibitem{Batrouni} Equilibrium And Non-equilibrium Statistical Thermodynamics. Michel Le Bellac, Fabrice Mortessagne
and G. George Batrouni. (2004)
	\bibitem{clock} Murty S.S. Challa and D.P. Landau Phys. Rev. B \textbf{33}, 437 (1986)
\end{thebibliography}

\end{document}